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Chapter 10: Q18P (page 496)

Using (10.19), show that ai aj =๐›ฟ i j.

Short Answer

Expert verified

The equation has been proven.

Step by step solution

01

Given Information:

The directions of the vectors ai, aj .

02

Definition of covariant basis vector:

A covariant vector, also known as a cotangent vector, has components that co-vary when the basis changes. That is, the components must be converted using the same matrix as the base matrix change.

03

Find the value:

The value of ai , aj are mentioned below.

Apply chain rule in the above equation, the equation becomes as follows.

Hence, the equation has been proven.

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Most popular questions from this chapter

Following what we did in equations (2.14) to (2.17), show that the direct product of a vector and a 3rd-rank tensor is a 4rh-rank tensor. Also show that the direct product of two 2nd-rank tensors is a 4rh-rank tensor. Generalize this to show that the direct product of two tensors of ranks m and n is a tensor of rank m + n .

Show that the nine quantities Tij=(โˆ‚Vi)/(โˆ‚xJ) (which are the Cartesian components of โˆ‡V where V is a vector) satisfy the transformation equations (2.14)for a Cartesian 2nd -rank tensor. Show that they do not satisfy the general tensor transformation equations as in (10.12) . Hint: Differentiate (10.9)or(10.10)partially with respect to, say,x'k. You should get the expected terms [as in(10.12) ] plus some extra terms; these extraneous terms show that(โˆ‚Vi)/(โˆ‚xJ) is not a tensor under general transformations. Comment: It is possible to express the components ofโˆ‡V correctly in general coordinate systems by taking into account the variation of the basis vectors in length and direction.

Parabolic cylinder.

:Do Problem 5 for the coordinate systems indicated in Problems 10 to 13.Eliptical cylinder.

Show that in a general coordinate system with variables x1, x2, x3, the contravariant basis vectors are given by

ai=โˆ‡xi=iโˆ‚xiโˆ‚x+jโˆ‚xiโˆ‚y+kโˆ‚xiโˆ‚z

Hint:Write the gradient in terms of its covariant components and the basis

vectors to getโˆ‡u=ajโˆ‚uโˆ‚xjand letu=xi .
See all solutions

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